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The Dons went into the clash at the Wham Stadium with a 1-0 advantage from last week's first leg on home soil.
However, that lead was eradicated in the 39th minute when Accrington's Josh Windass kept his cool from the penalty spot.
Then, in the 59th minute, it seemed that the hosts were destined for a crack at gaining promotion to League One as Piero Mingoia put them two goals to the good with a powerful shot.
Yet, Wimbledon frontman Adebayo Akinfenwa ensured that the tie would head into extra time when he sent a bullet header into the back of the home side's net.
Chances were at a premium during the early exchanges of the added 30 minutes, but that all changed 60 seconds before the interval as Wimbledon took a 3-2 aggregate lead.
Further drama then ensued during the second half as play was halted due to a floodlight failure.
It took several minutes for the action to get restarted, but once it did, Wimbledon were able to see out the minutes that remained to book their place at the home of English football against Plymouth Argyle.